Objects change their places?
The number highlighted below indicates that your objects share one action which means that they all share the same keyframes.



You can create an copy of the action for the active object by clicking on the number. After that you should be able to modify the object's keyframes individually.

Update:
If you want to affect multiple objects by a single action, I think the default approach would be to add an armature with a bone for every object, add a Child-Of constraint to each object, choosing the specific bone, and then creating an action which animates the bones.